Title:

Dynamics and statistical mechanics of point vortices in bounded domains

A general treatment of the dynamics and statistical mechanics of point vortices in bounded domains is introduced in Chapter 1. Chapter 2 then considers high positive energy statistical mechanics of 2D Euler vortices. In this case, the mostprobable equilibrium dynamics are given by solutions of the sinhPoisson equation and a particular heartshaped domain is found in which below a critical energy the solution has a dipolar structure and above it a monopolar structure. SinhPoisson predictions are compared to longtime averages of dynamical simulations of the $N$ vortex system in the same domain. Chapter 3 introduces a new algorithm (VORMFS) for the solution of generalised point vortex dynamics in an arbitrary domain. The algorithm only requires knowledge of the freespace Green's function and utilises the exponentially convergent method of fundamental solutions to obtain an approximation to the vortex Hamiltonian by solution of an appropriate boundary value problem. A number of test cases are presented, including quasigeostrophic shallow water (QGSW) point vortex motion (governed by a Bessel function). Chapter 4 concerns low energy (positive and negative) statistical mechanics of QGSW vortices in `Neumann oval' domains. In this case, the `vorticity fluctuation equation'  analogous to the sinhPoisson equation  is derived and solved to give expressions for key thermodynamic quantities. These theoretical expressions are compared with results from direct sampling of the microcanonical ensemble, using VORMFS to calculate the energy of the QGSW system. Chapter 5 considers the distribution of 2D Euler vortices in a Neumann oval. At high energies, vortices of one sign cluster in one lobe of the domain and vortices of the other sign cluster in the other lobe. For longtime simulations, these clusters are found to switch lobes. This behaviour is verified using results from the microcanonical ensemble.
